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Theorem chfacfscmulgsum 22209
Description: Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
Hypotheses
Ref Expression
chfacfisf.a 𝐴 = (𝑁 Mat 𝑅)
chfacfisf.b 𝐵 = (Base‘𝐴)
chfacfisf.p 𝑃 = (Poly1𝑅)
chfacfisf.y 𝑌 = (𝑁 Mat 𝑃)
chfacfisf.r × = (.r𝑌)
chfacfisf.s = (-g𝑌)
chfacfisf.0 0 = (0g𝑌)
chfacfisf.t 𝑇 = (𝑁 matToPolyMat 𝑅)
chfacfisf.g 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
chfacfscmulcl.x 𝑋 = (var1𝑅)
chfacfscmulcl.m · = ( ·𝑠𝑌)
chfacfscmulcl.e = (.g‘(mulGrp‘𝑃))
chfacfscmulgsum.p + = (+g𝑌)
Assertion
Ref Expression
chfacfscmulgsum (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑌   𝑛,𝑏   𝑛,𝑠,𝐵   0 ,𝑛   𝐵,𝑖,𝑠   𝑖,𝐺   𝑖,𝑀   𝑖,𝑁   𝑅,𝑖   𝑖,𝑋   𝑖,𝑌   ,𝑖   · ,𝑏,𝑖   𝑇,𝑛   ,𝑛   × ,𝑛   𝑖,𝑛
Allowed substitution hints:   𝐴(𝑖,𝑛,𝑠,𝑏)   𝐵(𝑏)   𝑃(𝑖,𝑛,𝑠,𝑏)   + (𝑖,𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑇(𝑖,𝑠,𝑏)   · (𝑛,𝑠)   × (𝑖,𝑠,𝑏)   (𝑛,𝑠,𝑏)   𝐺(𝑛,𝑠,𝑏)   𝑀(𝑠,𝑏)   (𝑖,𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑋(𝑛,𝑠,𝑏)   𝑌(𝑠,𝑏)   0 (𝑖,𝑠,𝑏)

Proof of Theorem chfacfscmulgsum
StepHypRef Expression
1 eqid 2736 . . 3 (Base‘𝑌) = (Base‘𝑌)
2 chfacfisf.0 . . 3 0 = (0g𝑌)
3 chfacfscmulgsum.p . . 3 + = (+g𝑌)
4 crngring 19976 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
54anim2i 617 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
653adant3 1132 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7 chfacfisf.p . . . . . . 7 𝑃 = (Poly1𝑅)
8 chfacfisf.y . . . . . . 7 𝑌 = (𝑁 Mat 𝑃)
97, 8pmatring 22041 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
106, 9syl 17 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
11 ringcmn 20003 . . . . 5 (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
1210, 11syl 17 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ CMnd)
1312adantr 481 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ CMnd)
14 nn0ex 12419 . . . 4 0 ∈ V
1514a1i 11 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ℕ0 ∈ V)
16 simpll 765 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
17 simplr 767 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))))
18 simpr 485 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
1916, 17, 183jca 1128 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ ℕ0))
20 chfacfisf.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
21 chfacfisf.b . . . . 5 𝐵 = (Base‘𝐴)
22 chfacfisf.r . . . . 5 × = (.r𝑌)
23 chfacfisf.s . . . . 5 = (-g𝑌)
24 chfacfisf.t . . . . 5 𝑇 = (𝑁 matToPolyMat 𝑅)
25 chfacfisf.g . . . . 5 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
26 chfacfscmulcl.x . . . . 5 𝑋 = (var1𝑅)
27 chfacfscmulcl.m . . . . 5 · = ( ·𝑠𝑌)
28 chfacfscmulcl.e . . . . 5 = (.g‘(mulGrp‘𝑃))
2920, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmulcl 22206 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
3019, 29syl 17 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
3120, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmulfsupp 22208 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖))) finSupp 0 )
32 nn0disj 13557 . . . 4 ((0...(𝑠 + 1)) ∩ (ℤ‘((𝑠 + 1) + 1))) = ∅
3332a1i 11 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((0...(𝑠 + 1)) ∩ (ℤ‘((𝑠 + 1) + 1))) = ∅)
34 nnnn0 12420 . . . . . 6 (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
35 peano2nn0 12453 . . . . . 6 (𝑠 ∈ ℕ0 → (𝑠 + 1) ∈ ℕ0)
3634, 35syl 17 . . . . 5 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ0)
37 nn0split 13556 . . . . 5 ((𝑠 + 1) ∈ ℕ0 → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
3836, 37syl 17 . . . 4 (𝑠 ∈ ℕ → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
3938ad2antrl 726 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
401, 2, 3, 13, 15, 30, 31, 33, 39gsumsplit2 19706 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖))))))
41 simpll 765 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
42 simplr 767 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))))
43 nncn 12161 . . . . . . . . . . . . 13 (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
44 add1p1 12404 . . . . . . . . . . . . 13 (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2))
4543, 44syl 17 . . . . . . . . . . . 12 (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2))
4645ad2antrl 726 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑠 + 1) + 1) = (𝑠 + 2))
4746fveq2d 6846 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (ℤ‘((𝑠 + 1) + 1)) = (ℤ‘(𝑠 + 2)))
4847eleq2d 2823 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↔ 𝑖 ∈ (ℤ‘(𝑠 + 2))))
4948biimpa 477 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → 𝑖 ∈ (ℤ‘(𝑠 + 2)))
5020, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmul0 22207 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ (ℤ‘(𝑠 + 2))) → ((𝑖 𝑋) · (𝐺𝑖)) = 0 )
5141, 42, 49, 50syl3anc 1371 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → ((𝑖 𝑋) · (𝐺𝑖)) = 0 )
5251mpteq2dva 5205 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖))) = (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 ))
5352oveq2d 7373 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )))
544, 9sylan2 593 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
55 ringmnd 19974 . . . . . . . . . 10 (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
5654, 55syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Mnd)
57563adant3 1132 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Mnd)
58 fvex 6855 . . . . . . . 8 (ℤ‘((𝑠 + 1) + 1)) ∈ V
5957, 58jctir 521 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V))
6059adantr 481 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V))
612gsumz 18646 . . . . . 6 ((𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
6260, 61syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
6353, 62eqtrd 2776 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = 0 )
6463oveq2d 7373 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖))))) = ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + 0 ))
65 fzfid 13878 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (0...(𝑠 + 1)) ∈ Fin)
66 elfznn0 13534 . . . . . . . 8 (𝑖 ∈ (0...(𝑠 + 1)) → 𝑖 ∈ ℕ0)
6766, 19sylan2 593 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ ℕ0))
6867, 29syl 17 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
6968ralrimiva 3143 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ∀𝑖 ∈ (0...(𝑠 + 1))((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
701, 13, 65, 69gsummptcl 19744 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) ∈ (Base‘𝑌))
711, 3, 2mndrid 18577 . . . 4 ((𝑌 ∈ Mnd ∧ (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) ∈ (Base‘𝑌)) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + 0 ) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
7257, 70, 71syl2an2r 683 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + 0 ) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
7364, 72eqtrd 2776 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖))))) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
7434ad2antrl 726 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ ℕ0)
751, 3, 13, 74, 68gsummptfzsplit 19709 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 𝑋) · (𝐺𝑖))))))
76 elfznn0 13534 . . . . . . 7 (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0)
7776, 30sylan2 593 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
781, 3, 13, 74, 77gsummptfzsplitl 19710 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 𝑋) · (𝐺𝑖))))))
7957adantr 481 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Mnd)
80 0nn0 12428 . . . . . . . 8 0 ∈ ℕ0
8180a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 0 ∈ ℕ0)
8220, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmulcl 22206 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 0 ∈ ℕ0) → ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
8381, 82mpd3an3 1462 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
84 oveq1 7364 . . . . . . . . 9 (𝑖 = 0 → (𝑖 𝑋) = (0 𝑋))
85 fveq2 6842 . . . . . . . . 9 (𝑖 = 0 → (𝐺𝑖) = (𝐺‘0))
8684, 85oveq12d 7375 . . . . . . . 8 (𝑖 = 0 → ((𝑖 𝑋) · (𝐺𝑖)) = ((0 𝑋) · (𝐺‘0)))
871, 86gsumsn 19731 . . . . . . 7 ((𝑌 ∈ Mnd ∧ 0 ∈ ℕ0 ∧ ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌)) → (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((0 𝑋) · (𝐺‘0)))
8879, 81, 83, 87syl3anc 1371 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((0 𝑋) · (𝐺‘0)))
8988oveq2d 7373 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 𝑋) · (𝐺𝑖))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))))
9078, 89eqtrd 2776 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))))
91 ovexd 7392 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑠 + 1) ∈ V)
92 1nn0 12429 . . . . . . . 8 1 ∈ ℕ0
9392a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 1 ∈ ℕ0)
9474, 93nn0addcld 12477 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
9520, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmulcl 22206 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ (𝑠 + 1) ∈ ℕ0) → (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
9694, 95mpd3an3 1462 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
97 oveq1 7364 . . . . . . 7 (𝑖 = (𝑠 + 1) → (𝑖 𝑋) = ((𝑠 + 1) 𝑋))
98 fveq2 6842 . . . . . . 7 (𝑖 = (𝑠 + 1) → (𝐺𝑖) = (𝐺‘(𝑠 + 1)))
9997, 98oveq12d 7375 . . . . . 6 (𝑖 = (𝑠 + 1) → ((𝑖 𝑋) · (𝐺𝑖)) = (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))
1001, 99gsumsn 19731 . . . . 5 ((𝑌 ∈ Mnd ∧ (𝑠 + 1) ∈ V ∧ (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))
10179, 91, 96, 100syl3anc 1371 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))
10290, 101oveq12d 7375 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 𝑋) · (𝐺𝑖))))) = (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))))
103 fzfid 13878 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (1...𝑠) ∈ Fin)
104 simpll 765 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
105 simplr 767 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))))
106 elfznn 13470 . . . . . . . . . 10 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
107106nnnn0d 12473 . . . . . . . . 9 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ0)
108107adantl 482 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ0)
109104, 105, 108, 29syl3anc 1371 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
110109ralrimiva 3143 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
1111, 13, 103, 110gsummptcl 19744 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) ∈ (Base‘𝑌))
1121, 3mndass 18565 . . . . 5 ((𝑌 ∈ Mnd ∧ ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) ∈ (Base‘𝑌) ∧ ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))))
11379, 111, 83, 96, 112syl13anc 1372 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))))
114106nnne0d 12203 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑠) → 𝑖 ≠ 0)
115114ad2antlr 725 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ 0)
116 neeq1 3006 . . . . . . . . . . . . . 14 (𝑛 = 𝑖 → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
117116adantl 482 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
118115, 117mpbird 256 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ 0)
119 eqneqall 2954 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑛 ≠ 0 → 0 = (𝑇‘(𝑏‘(𝑖 − 1)))))
120118, 119mpan9 507 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = (𝑇‘(𝑏‘(𝑖 − 1))))
121 simplr 767 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 𝑛 = 𝑖)
122 eqeq1 2740 . . . . . . . . . . . . . . . . 17 (0 = 𝑛 → (0 = 𝑖𝑛 = 𝑖))
123122eqcoms 2744 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → (0 = 𝑖𝑛 = 𝑖))
124123adantl 482 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (0 = 𝑖𝑛 = 𝑖))
125121, 124mpbird 256 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = 𝑖)
126125fveq2d 6846 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑏‘0) = (𝑏𝑖))
127126fveq2d 6846 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑇‘(𝑏‘0)) = (𝑇‘(𝑏𝑖)))
128127oveq2d 7373 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) = ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))
129120, 128oveq12d 7375 . . . . . . . . . 10 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
130 elfz2 13431 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) ↔ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)))
131 zleltp1 12554 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
132131ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
1331323adant1 1130 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
134133biimpcd 248 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖𝑠 → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
135134adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((1 ≤ 𝑖𝑖𝑠) → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
136135impcom 408 . . . . . . . . . . . . . . . . . . . 20 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → 𝑖 < (𝑠 + 1))
137136orcd 871 . . . . . . . . . . . . . . . . . . 19 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖))
138 zre 12503 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ ℤ → 𝑠 ∈ ℝ)
139 1red 11156 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ ℤ → 1 ∈ ℝ)
140138, 139readdcld 11184 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 ∈ ℤ → (𝑠 + 1) ∈ ℝ)
141 zre 12503 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
142140, 141anim12ci 614 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
1431423adant1 1130 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
144 lttri2 11237 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
145143, 144syl 17 . . . . . . . . . . . . . . . . . . . 20 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
146145adantr 481 . . . . . . . . . . . . . . . . . . 19 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
147137, 146mpbird 256 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → 𝑖 ≠ (𝑠 + 1))
148130, 147sylbi 216 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑠) → 𝑖 ≠ (𝑠 + 1))
149148ad2antlr 725 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ (𝑠 + 1))
150 neeq1 3006 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑖 → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
151150adantl 482 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
152149, 151mpbird 256 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ (𝑠 + 1))
153152adantr 481 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ (𝑠 + 1))
154153neneqd 2948 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = (𝑠 + 1))
155154pm2.21d 121 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → (𝑛 = (𝑠 + 1) → (𝑇‘(𝑏𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
156155imp 407 . . . . . . . . . . 11 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
157106nnred 12168 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℝ)
158 eleq1w 2820 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑖 → (𝑛 ∈ ℝ ↔ 𝑖 ∈ ℝ))
159157, 158syl5ibrcom 246 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) → (𝑛 = 𝑖𝑛 ∈ ℝ))
160159adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑛 = 𝑖𝑛 ∈ ℝ))
161160imp 407 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ∈ ℝ)
16274nn0red 12474 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ ℝ)
163162ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑠 ∈ ℝ)
164 1red 11156 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 1 ∈ ℝ)
165163, 164readdcld 11184 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑠 + 1) ∈ ℝ)
166130, 136sylbi 216 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) → 𝑖 < (𝑠 + 1))
167166ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 < (𝑠 + 1))
168 breq1 5108 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑖 → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
169168adantl 482 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
170167, 169mpbird 256 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 < (𝑠 + 1))
171161, 165, 170ltnsymd 11304 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ¬ (𝑠 + 1) < 𝑛)
172171pm2.21d 121 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ((𝑠 + 1) < 𝑛0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
173172ad2antrr 724 . . . . . . . . . . . . 13 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → ((𝑠 + 1) < 𝑛0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
174173imp 407 . . . . . . . . . . . 12 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
175 simp-4r 782 . . . . . . . . . . . . . . 15 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 = 𝑖)
176175fvoveq1d 7379 . . . . . . . . . . . . . 14 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑖 − 1)))
177176fveq2d 6846 . . . . . . . . . . . . 13 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
178175fveq2d 6846 . . . . . . . . . . . . . . 15 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏𝑛) = (𝑏𝑖))
179178fveq2d 6846 . . . . . . . . . . . . . 14 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏𝑛)) = (𝑇‘(𝑏𝑖)))
180179oveq2d 7373 . . . . . . . . . . . . 13 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇𝑀) × (𝑇‘(𝑏𝑛))) = ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))
181177, 180oveq12d 7375 . . . . . . . . . . . 12 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
182174, 181ifeqda 4522 . . . . . . . . . . 11 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
183156, 182ifeqda 4522 . . . . . . . . . 10 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
184129, 183ifeqda 4522 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
185 ovexd 7392 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))) ∈ V)
18625, 184, 108, 185fvmptd2 6956 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝐺𝑖) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
187186oveq2d 7373 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 𝑋) · (𝐺𝑖)) = ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
188187mpteq2dva 5205 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))))
189188oveq2d 7373 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))))
190 nn0p1gt0 12442 . . . . . . . . . . . . . 14 (𝑠 ∈ ℕ0 → 0 < (𝑠 + 1))
191 0red 11158 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ℕ0 → 0 ∈ ℝ)
192 ltne 11252 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
193191, 192sylan 580 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ0 ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
194 neeq1 3006 . . . . . . . . . . . . . . 15 (𝑛 = (𝑠 + 1) → (𝑛 ≠ 0 ↔ (𝑠 + 1) ≠ 0))
195193, 194syl5ibrcom 246 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℕ0 ∧ 0 < (𝑠 + 1)) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
19634, 190, 195syl2anc2 585 . . . . . . . . . . . . 13 (𝑠 ∈ ℕ → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
197196ad2antrl 726 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
198197imp 407 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 𝑛 ≠ 0)
199 eqneqall 2954 . . . . . . . . . . 11 (𝑛 = 0 → (𝑛 ≠ 0 → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏𝑠))))
200198, 199mpan9 507 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ 𝑛 = 0) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏𝑠)))
201 iftrue 4492 . . . . . . . . . . 11 (𝑛 = (𝑠 + 1) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = (𝑇‘(𝑏𝑠)))
202201ad2antlr 725 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = (𝑇‘(𝑏𝑠)))
203200, 202ifeqda 4522 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))) = (𝑇‘(𝑏𝑠)))
20474, 35syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
205 fvexd 6857 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇‘(𝑏𝑠)) ∈ V)
20625, 203, 204, 205fvmptd2 6956 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝐺‘(𝑠 + 1)) = (𝑇‘(𝑏𝑠)))
207206oveq2d 7373 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) = (((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))))
20843ad2ant2 1134 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
209 eqid 2736 . . . . . . . . . . . . . 14 (Base‘𝑃) = (Base‘𝑃)
21026, 7, 209vr1cl 21588 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
211208, 210syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑋 ∈ (Base‘𝑃))
212 eqid 2736 . . . . . . . . . . . . . 14 (mulGrp‘𝑃) = (mulGrp‘𝑃)
213212, 209mgpbas 19902 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
214 eqid 2736 . . . . . . . . . . . . . 14 (1r𝑃) = (1r𝑃)
215212, 214ringidval 19915 . . . . . . . . . . . . 13 (1r𝑃) = (0g‘(mulGrp‘𝑃))
216213, 215, 28mulg0 18879 . . . . . . . . . . . 12 (𝑋 ∈ (Base‘𝑃) → (0 𝑋) = (1r𝑃))
217211, 216syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0 𝑋) = (1r𝑃))
2187ply1crng 21569 . . . . . . . . . . . . . . 15 (𝑅 ∈ CRing → 𝑃 ∈ CRing)
219218anim2i 617 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
2202193adant3 1132 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
2218matsca2 21769 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
222220, 221syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑃 = (Scalar‘𝑌))
223222fveq2d 6846 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (1r𝑃) = (1r‘(Scalar‘𝑌)))
224217, 223eqtrd 2776 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0 𝑋) = (1r‘(Scalar‘𝑌)))
225224adantr 481 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (0 𝑋) = (1r‘(Scalar‘𝑌)))
226225oveq1d 7372 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((0 𝑋) · (𝐺‘0)) = ((1r‘(Scalar‘𝑌)) · (𝐺‘0)))
2277, 8pmatlmod 22042 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod)
2284, 227sylan2 593 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
2292283adant3 1132 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ LMod)
23020, 21, 7, 8, 22, 23, 2, 24, 25chfacfisf 22203 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
2314, 230syl3anl2 1413 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
232231, 81ffvelcdmd 7036 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝐺‘0) ∈ (Base‘𝑌))
233 eqid 2736 . . . . . . . . . 10 (Scalar‘𝑌) = (Scalar‘𝑌)
234 eqid 2736 . . . . . . . . . 10 (1r‘(Scalar‘𝑌)) = (1r‘(Scalar‘𝑌))
2351, 233, 27, 234lmodvs1 20350 . . . . . . . . 9 ((𝑌 ∈ LMod ∧ (𝐺‘0) ∈ (Base‘𝑌)) → ((1r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
236229, 232, 235syl2an2r 683 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((1r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
237 iftrue 4492 . . . . . . . . 9 (𝑛 = 0 → if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
238 ovexd 7392 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) ∈ V)
23925, 237, 81, 238fvmptd3 6971 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝐺‘0) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
240226, 236, 2393eqtrd 2780 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((0 𝑋) · (𝐺‘0)) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
241207, 240oveq12d 7375 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 𝑋) · (𝐺‘0))) = ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
2421, 3cmncom 19580 . . . . . . 7 ((𝑌 ∈ CMnd ∧ ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 𝑋) · (𝐺‘0))))
24313, 83, 96, 242syl3anc 1371 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 𝑋) · (𝐺‘0))))
244 ringgrp 19969 . . . . . . . . 9 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
24510, 244syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Grp)
246245adantr 481 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Grp)
247207, 96eqeltrrd 2839 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌))
24810adantr 481 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Ring)
24924, 20, 21, 7, 8mat2pmatbas 22075 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
2504, 249syl3an2 1164 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
251250adantr 481 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇𝑀) ∈ (Base‘𝑌))
252 simpl1 1191 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑁 ∈ Fin)
253208adantr 481 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑅 ∈ Ring)
254 elmapi 8787 . . . . . . . . . . . 12 (𝑏 ∈ (𝐵m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
255254adantl 482 . . . . . . . . . . 11 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
256255adantl 482 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
257 0elfz 13538 . . . . . . . . . . . 12 (𝑠 ∈ ℕ0 → 0 ∈ (0...𝑠))
25834, 257syl 17 . . . . . . . . . . 11 (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
259258ad2antrl 726 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 0 ∈ (0...𝑠))
260256, 259ffvelcdmd 7036 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
26124, 20, 21, 7, 8mat2pmatbas 22075 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
262252, 253, 260, 261syl3anc 1371 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
2631, 22ringcl 19981 . . . . . . . 8 ((𝑌 ∈ Ring ∧ (𝑇𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
264248, 251, 262, 263syl3anc 1371 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
2651, 2, 23, 3grpsubadd0sub 18834 . . . . . . 7 ((𝑌 ∈ Grp ∧ (((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌) ∧ ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
266246, 247, 264, 265syl3anc 1371 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
267241, 243, 2663eqtr4d 2786 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
268189, 267oveq12d 7375 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
269113, 268eqtrd 2776 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
27075, 102, 2693eqtrd 2780 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
27140, 73, 2703eqtrd 2780 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  cun 3908  cin 3909  c0 4282  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556  Grpcgrp 18748  -gcsg 18750  .gcmg 18872  CMndccmn 19562  mulGrpcmgp 19896  1rcur 19913  Ringcrg 19964  CRingccrg 19965  LModclmod 20322  var1cv1 21547  Poly1cpl1 21548   Mat cmat 21754   matToPolyMat cmat2pmat 22053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-mamu 21733  df-mat 21755  df-mat2pmat 22056
This theorem is referenced by:  cpmadugsum  22227
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